Properties of Identity, Inverses, and Zero

By the end of this section, you will be able to: recognize the identity properties of addition and multiplication, use the inverse properties of addition and multiplication, use the properties of zero, and simplify expressions using the properties of identities, inverses, and zero.

Recognize the identity properties of addition and multiplication

What happens when we add zero to any number? Adding zero doesn’t change the value. For this reason, we call $0$ the additive identity. For example,

13 + 0 = 13 \qquad -14 + 0 = -14 \qquad 0 + (-3x) = -3x

What happens when you multiply any number by one? Multiplying by one doesn’t change the value. So we call $1$ the multiplicative identity. For example,

43 \cdot 1 = 43 \qquad -27 \cdot 1 = -27 \qquad 1 \cdot \tfrac{6y}{5} = \tfrac{6y}{5}

Identity properties.

The identity property of addition: for any real number $a$ ,

a + 0 = a \qquad\qquad 0 + a = a

$0$ is called the additive identity.

The identity property of multiplication: for any real number $a$ ,

a \cdot 1 = a \qquad\qquad 1 \cdot a = a

$1$ is called the multiplicative identity.

Example. Identify whether each equation demonstrates the identity property of addition or multiplication: (a) $7 + 0 = 7$ (b) $-16(1) = -16$ .

(a) We are adding $0$ , so this uses the identity property of addition.

(b) We are multiplying by $1$ , so this uses the identity property of multiplication.

How many of these two equations demonstrate the identity property of addition (rather than multiplication): $23 + 0 = 23$ , and $-37(1) = -37$ ?

Use the inverse properties of addition and multiplication

What number added to $5$ gives the additive identity, $0$ ? We know $5 + (-5) = 0$ . What number added to $-6$ gives $0$ ? We know $-6 + 6 = 0$ . Notice that in each case, the missing number was the opposite of the number.

We call $-a$ the additive inverse of $a$ . The opposite of a number is its additive inverse. A number and its opposite add to $0$ , which is the additive identity.

What number multiplied by $\tfrac{2}{3}$ gives the multiplicative identity, $1$ ? We know $\tfrac{2}{3} \cdot \tfrac{3}{2} = 1$ . What number multiplied by $2$ gives $1$ ? We know $2 \cdot \tfrac{1}{2} = 1$ . Notice that in each case, the missing number was the reciprocal of the number.

We call $\tfrac{1}{a}$ the multiplicative inverse of $a$ (where $a \neq 0$ ). The reciprocal of a number is its multiplicative inverse. A number and its reciprocal multiply to $1$ , which is the multiplicative identity.

Inverse properties.

Inverse Property of Addition: for any real number $a$ ,

a + (-a) = 0

$-a$ is the additive inverse of $a$ .

Inverse Property of Multiplication: for any real number $a \neq 0$ ,

a \cdot \tfrac{1}{a} = 1

$\tfrac{1}{a}$ is the multiplicative inverse of $a$ .

Example. Find the additive inverse of each expression: (a) $13$ (b) $-\tfrac{5}{8}$ (c) $0.6$ .

To find the additive inverse, we find the opposite: (a) the additive inverse of $13$ is $-13$ (b) the additive inverse of $-\tfrac{5}{8}$ is $\tfrac{5}{8}$ (c) the additive inverse of $0.6$ is $-0.6$ .

Find the additive inverse of $18$ .

Find the additive inverse of $1.2$ .

Example. Find the multiplicative inverse: (a) $9$ (b) $-\tfrac{1}{9}$ (c) $0.9$ .

To find the multiplicative inverse, we find the reciprocal: (a) the multiplicative inverse of $9$ is $\tfrac{1}{9}$ (b) the multiplicative inverse of $-\tfrac{1}{9}$ is $-9$ (c) to find the multiplicative inverse of $0.9$ , we first convert it to a fraction, $\tfrac{9}{10}$ , then find the reciprocal, $\tfrac{10}{9}$ .

Find the multiplicative inverse of $5$ .

Find the multiplicative inverse of $0.3$ .

Use the properties of zero

We have already learned that zero is the additive identity, since it can be added to any number without changing the number’s identity. But zero also has some special properties when it comes to multiplication and division.

Multiplication by zero

What happens when you multiply a number by $0$ ? Multiplying by $0$ makes the product equal zero. The product of any real number and $0$ is $0$ .

Multiplication by zero. For any real number $a$ ,

a \cdot 0 = 0 \qquad\qquad 0 \cdot a = 0

Example. Simplify: (a) $-8 \cdot 0$ (b) $\tfrac{5}{12} \cdot 0$ (c) $0(2.94)$ .

The product of any real number and $0$ is $0$ , so all three expressions simplify to $0$ .

Simplify: $-14 \cdot 0$ .

Dividing with zero

What about dividing with $0$ ? Think about a real example: if there are no cookies in the cookie jar and three people want to share them, how many cookies would each person get? There are $0$ cookies to share, so each person gets $0$ cookies:

0 \div 3 = 0 \text{ because } 0 \cdot 3 = 0

Division of zero. For any real number

a

, except

0

\tfrac{0}{a} = 0

and

0 \div a = 0

Zero divided by any real number except zero is zero.

Example. Simplify: (a) $0 \div 5$ (b) $\tfrac{0}{-2}$ (c) $0 \div \tfrac{7}{8}$ .

Zero divided by any real number, except $0$ , is zero — so all three expressions simplify to $0$ .

Simplify: $0 \div 11$ .

Now let’s think about dividing a number by zero. What is the result of dividing $4$ by $0$ ? Think about the related multiplication fact — is there a number that multiplied by $0$ gives $4$ ? Since any real number multiplied by $0$ equals $0$ , there is no real number that can be multiplied by $0$ to obtain $4$ . We can conclude that there is no answer to $4 \div 0$ , and so we say that division by zero is undefined.

Division by zero. For any real number

a

\tfrac{a}{0}

and

a \div 0

are undefined.

Division by zero is undefined.

Example. Simplify: (a) $7.5 \div 0$ (b) $\tfrac{-32}{0}$ (c) $\tfrac{4}{9} \div 0$ .

Division by zero is undefined, so all three expressions are undefined.

How many of these four expressions are undefined: $0 \div 11$ , $16.4 \div 0$ , $\tfrac{-2}{0}$ , and $\tfrac{0}{-6}$ ?

We summarize the properties of zero:

Properties of zero.

Multiplication by Zero: for any real number $a$ , $a \cdot 0 = 0$ and $0 \cdot a = 0$ — the product of any number and $0$ is $0$ .

Division by Zero: for any real number $a \neq 0$ , $\tfrac{0}{a} = 0$ (zero divided by any real number, except itself, is zero), but $\tfrac{a}{0}$ is undefined (division by zero is undefined).

Simplify expressions using the properties of identities, inverses, and zero

We will now practice using the properties of identities, inverses, and zero to simplify expressions.

Example. Simplify: $3x + 15 - 3x$ .

Notice the additive inverses, $3x$ and $-3x$ ; they combine to $0$ , leaving the identity:

3x + 15 - 3x = (3x - 3x) + 15 = 0 + 15 = 15

Simplify: $-12z + 9 + 12z$ .

Example. Simplify: $4(0.25q)$ .

Regroup using the associative property, then multiply — the two factors are reciprocals, so the coefficient becomes the multiplicative identity:

4(0.25q) = [4(0.25)]q = 1.00q = q

Simplify: $2(0.5p)$ .

Example. Simplify: $\tfrac{0}{n+5}$ , where $n \neq -5$ .

Zero divided by any real number except itself is zero:

\tfrac{0}{n+5} = 0

Simplify: $\tfrac{0}{m + 7}$ , where $m \ne -7$ .

Example. Simplify: $\tfrac{10 - 3p}{0}$ .

Division by zero is undefined, so this expression is undefined — no matter what $p$ is, the denominator is always $0$ .

Example. Simplify: $\tfrac{3}{4} \cdot \tfrac{4}{3}(6x + 12)$ .

We cannot combine the terms in parentheses, so we multiply the two fractions first — they are reciprocals, so their product is the multiplicative identity:

\tfrac{3}{4} \cdot \tfrac{4}{3}(6x + 12) = 1(6x + 12) = 6x + 12

Simplify: $\tfrac{2}{5} \cdot \tfrac{5}{2}(20y + 50)$ .

All the properties of real numbers used in this chapter are summarized below:

Property	Of Addition	Of Multiplication
Commutative Property — if $a$ and $b$ are real numbers, then…	$a + b = b + a$	$a \cdot b = b \cdot a$
Associative Property — if $a$ , $b$ , $c$ are real numbers, then…	$(a + b) + c = a + (b + c)$	$(a \cdot b) \cdot c = a \cdot (b \cdot c)$
Identity Property	$0$ is the additive identity: $a + 0 = a$ , $0 + a = a$	$1$ is the multiplicative identity: $a \cdot 1 = a$ , $1 \cdot a = a$
Inverse Property	$-a$ is the additive inverse of $a$ : $a + (-a) = 0$	for $a \neq 0$ , $\tfrac{1}{a}$ is the multiplicative inverse of $a$ : $a \cdot \tfrac{1}{a} = 1$
Properties of Zero	$a \cdot 0 = 0$ , $0 \cdot a = 0$	for $a \neq 0$ : $\tfrac{0}{a} = 0$ ; $\tfrac{a}{0}$ is undefined

Distributive Property: if $a$ , $b$ , $c$ are real numbers, then $a(b + c) = ab + ac$ .

Key terms

additive identity — $0$ ; adding it to any number leaves the number unchanged. multiplicative identity — $1$ ; multiplying by it leaves any number unchanged. additive inverse — the opposite of a number; a number and its additive inverse sum to $0$ . multiplicative inverse — the reciprocal of a number; a nonzero number and its multiplicative inverse multiply to $1$ . Zero times any number is $0$ ; zero divided by any nonzero number is $0$ ; but division by zero is undefined.

This section is adapted from Prealgebra 2e, Section 7.4: Properties of Identity, Inverses, and Zero by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recreated the chapter-wide properties summary (Table 7.1) as a markdown table; omitted the Be Prepared quiz, media links, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback, rephrasing the identity-property classification problem and one division-by-zero problem as counting questions so they can be graded as math expressions.

Distributive Property Systems of Measurement